Trigonometric Identities

cscθ

secθ

tanθ

cotθ

sin²θ

cos²θ

1

cot²θ

csc²θ

sec²θ

tan²θ

sinθ

1/cscθ

cosθ

1/secθ

secθ reciprocal identity

1/cosθ

cscθ reciprocal identity

1/sinθ

cotθ reciprocal identity

1/tanθ

tanθ quotient identity

sinθ/cosθ

cotθ quotient identity

cosθ/sinθ

-sinθ odd-even identity

sin(-θ)

cosθ odd-even identity

cos(-θ)

-tanθ odd-even identity

tan(-θ)

-cscθ odd-even identity

csc(-θ)

secθ odd-even identity

sec(-θ)

-cotθ odd-even identity

cot(-θ)

sinθ cofunction identity

cos (π/2-θ)

cosθ cofunction identity

sin (π/2-θ)

tanθ cofunction identity

cot (π/2-θ)

cscθ cofunction identity

sec (π/2-θ)

secθ cofunction identity

csc (π/2-θ)

cotθ cofunction identity

tan (π/2-θ)

sin²θ + cos²θ Pythagorean identity

1

1 + tan²θ Pythagorean identity

sec²θ

1 + cot²θ Pythagorean identity

csc²θ

sin(α + β) sum identity

sinαcosβ + cosαsinβ

sin(α - β) difference identity

sinαcosβ - cosαsinβ

cos(α + β) sum identity

cosαcosβ - sinαsinβ

cos(α - β) difference identity

cosαcosβ + sinαsinβ

tan(α + β) sum identity

tanα + tanβ/1 - tanαtanβ

tan(α - β) difference identity

tanα - tanβ/1 + tanαtanβ

sin 2θ double-angle identity

2 sinθcosθ

cos 2θ double-angle identity

cos²θ - sin²θ

2 cos²θ - 1

1 - 2 sin²θ

tan 2θ double-angle identity

2 tanθ/1 - tan²θ

sin½θ half-angle identity

±√1 - cosθ/2

cos½θ half-angle identity

±√1 + cosθ/2

tan½θ half-angle identity

±√1 - cosθ/1 + cosθ

1/sin x

csc x

1/cos x

sec x

1/tan x

cot x

1/csc x

sin x

1/sec x

cos x

1/cot x

tan x

sin²x+cos²x

1

1-cos²x

sin²x

1-sin²x

cos²x

1+tan²x

sec²x

1+cot²x

csc²x

sec²x-1

tan²x

csc²x-1

cot²x

sin(-x)

-sin x

csc(-x)

-csc x

cos(-x)

cos x

sec(-x)

sec x

tan(-x)

-tan x

cot(-x)

-cot x

sin(π/2-x)

cos x

cos(π/2-x)

sin x

tan(π/2-x)

cot x

csc(π/2-x)

sec x

sec(π/2-x)

csc x

cot(π/2-x)

tan x

sin x / cos x

tan x

cos x/ sin x

cot x

Identity (math)

A mathematical statement or equation that is true for any value.

Pythagorean Identities

A trigonometric identity that is derived using the Pythagorean Theorem.

a² + b² = c²

x² + y² = r²

sin²θ + cos²θ = 1 (y=sin, x=cos, r=1 on unit circle)

1 + cot²θ = csc²θ (divide everything by sin²θ)

tan²θ + 1 = sec²θ (divide everything by cos²θ)

Quotient

The answer to a division problem.

Quotient Identities

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Reciprocal

One of a pair of numbers whose product is 1.

Reciprocal Identities

Expression

A mathematical statement (variables,numbers, and operation symbols) that does not contain an equal sign.

Equation

A mathematical statement that two expressions are equal.

Verify

To confirm or prove something is true.

Simplify

To make something easier or less complicated.

Solve

To find an answer to or explanation for a problem.

Evaluate

For a specific input, find the numerical output of an function. Plug a number in to get a solution.

Isolate

To get a variable alone in a mathematical equation.

Factor (verb)

To find the factors of a number or expression.

Factor (noun)

A number which divides into another number exactly without leaving a remainder.

Distribute (Expand)

Multiplying each term inside the parentheses with the term outside of the parentheses.

Combine Like Terms

Simplifying an expression by using mathematical operations of terms with the same variables and exponents.

Coefficient

A number multiplied by a variable in an algebraic expression.

Variable

A letter or symbol used to represent numbers in Algebra.

Constant

A value that does not change.

Radical

Another name for a root: one might refer to √2 as "radical 2."

Undefined

Having no solution.

(Examples: dividing by zero, negative square roots, the slope of a vertical line)

Domain

All the possible x-values that can be input into a function.

Range

All the possible y-values that can be the output of a function.

Inverse Function

The function obtained by switching the domain (x) and range (y) in a function.